#124702

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,7,8,5)(4,6)(9,10,11,14,12,13,15,16)(17,18)(19,29,32,46)(20,25)(21,35)(22,27)(23,28)(24,31,40,47)(26,34)(30,45,33,43)(36,41,48,39)(37,44)(38,42);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,30)(21,36)(22,33)(23,32)(25,39)(26,43)(27,41)(28,40)(29,42)(31,37)(34,47)(35,46)(38,48)(44,45);(10,13)(14,16)(17,20,23,22)(18,25,28,27)(19,30,32,33)(21,34,38,37)(24,39,40,41)(26,42,44,35)(29,45,46,43)(31,36,47,48)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31898);